Intersection between the solid and a plane perpendicular

In summary, the conversation discusses finding the volume of a solid with a circular base of radius r=4, given different intersections with a plane perpendicular to a set diameter. The first case involves a square, and the integration is given as 4 times the integral of (r^2-x^2) from -4 to 4. The second case involves an isosceles right triangle with one side on the circular base, and the area can be found using Pythagoras' theorem. The conversation concludes by discussing different ways to draw the triangle and finding the volume from there.
  • #1
Telemachus
835
30

Homework Statement


A solid has a circular base of radius r=4. Find in each case the volume of the solid if all intersection between the solid and a plane perpendicular to a set diameter is:

a) A square
b) A triangle rectangle isosceles which one of its sides is on the circular base.

The Attempt at a Solution



The first I thought was in some kind of cylindroid. I know I must find an equation that I can integrate. So, the sides of the squares will be given by:

[tex]z=2\sqrt{r^2-x^2}[/tex]

Is the integration given by: [tex]\displaystyle\int_{-4}^{4}[2\sqrt{r^2-x^2}]^2dx=4\displaystyle\int_{-4}^{4}(r^2-x^2)dx[/tex] for the square?

And how can I find the volume of the other solid b)?
 
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  • #2


That looks fine to me. Ok, so the area of a square with side z is z^2. And you integrated the area to get the volume. Fine. What's the area of a isosceles right triangle with side z? I'm not sure why this is confusing you. Are you worried about Odysseus' return?
 
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  • #3


Haha I still drunk from Blooms day :p

The problem I got is... the area of the triangle: [tex]\diplaystyle\frac{bh}{2}[/tex]
How do I get h? Pythagoras?
 
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  • #4


Telemachus said:
Haha I still drunk from Blooms day :p

The problem I got is... the area of the triangle: [tex]diplayfrac\style{bh}{2}[/tex]
How do I get h? Pythagoras?

In the triangle I'm thinking of both b and h are z. In an isosceles right triangle both sides are equal, aren't they?
 
  • #5


If both of them are equal one each other, then they are not equal to the base. So I must find h...
 
  • #6


Telemachus said:
If both of them are equal one each other, then they are not equal to the base. So I must find h...

There are two different ways to draw the picture. The question says 'one of its sides is on the base'. I took this to mean that it isn't the hypotenuse that has length z, but rather one of the legs. I'm probably wrong. If I am, yes, use pythagoras to find the missing length.
 
  • #7


Sorry, I didn't give a righteous interpretation at what you said in the first place. You're right, it's easier to think of both, the base and the high as the information I already get to. So, it won't be hard to find the volume from there.

Thanks Dick.
 

FAQ: Intersection between the solid and a plane perpendicular

What is the definition of the intersection between a solid and a plane perpendicular?

The intersection between a solid and a plane perpendicular is the set of all points where the solid intersects the plane at a 90-degree angle, creating a cross-section of the solid.

How is the intersection between a solid and a plane perpendicular represented mathematically?

The intersection between a solid and a plane perpendicular can be represented mathematically using equations of the solid and plane. The equations can be solved simultaneously to find the coordinates of the points in the intersection.

Can the intersection between a solid and a plane perpendicular be a closed shape?

Yes, the intersection between a solid and a plane perpendicular can be a closed shape, depending on the shape of the solid and the orientation of the plane. For example, if the solid is a cube and the plane intersects one of its faces, the resulting shape will be a square, which is a closed shape.

What is the significance of studying the intersection between a solid and a plane perpendicular?

Studying the intersection between a solid and a plane perpendicular can help us understand the properties of the solid, such as its dimensions and cross-sectional area. It is also useful in applications such as engineering and architecture, where the intersection can be used to create 3-dimensional structures.

How does the angle of the plane affect the intersection between a solid and a plane perpendicular?

The angle of the plane can greatly impact the shape and size of the intersection between a solid and a plane perpendicular. For example, if the plane is parallel to one of the faces of a cube, the resulting intersection will be a rectangle with the same dimensions as the face. However, if the plane is at an angle, the resulting intersection will be a different shape with different dimensions.

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